Constructions are given of different kinds of flats in the projective space which are external to the Grassmannian of lines of PG(4,2). In particular it is shown that there exist precisely two GL(5,2)-orbits of external 4-flats, each with stabilizer group 31:5. (No 5-flat is external.) For each k=1,2,3, two distinct kinds of external k-flats are simply constructed out of certain partial spreads in PG(4,2) of size k+2. A third kind of external plane, with stabilizer 23:(7:3), is also shown to exist. With the aid of a certain key counting lemma, it is proved that the foregoing amounts to a complete classification of external flats.@en

We consider the following `even hyperplane construction¿ of flats in the projective space which are external to the Grassmannian of lines of PG(4,2). Let the Grassmann image in of a partial spread in PG(4,2) be . Then is an r-cap on . Using the recent classification [N.A. Gordon, R. Shaw, L.H. Soicher, Classification of Partial Spreads in PG(4,2), pp. 63, available from: http://www.hull.ac.uk/maths/people/rs/staffdetails.html] of partial spreads in PG(4,2), we determine those partial spreads such that the projective space generated by the points mijmi+mj is an external flat. We show that, in this simple manner, we may construct seven out of the ten GL(5,2)-orbits of external flats. Keywords: Grassmannian G1,4,2; Partial spreads; External flats; GL(5,2)-orbits@en